Answer:To find the value of (1 + i)⁸, we can use the binomial theorem. The binomial theorem allows us to expand expressions of the form (a + b)ⁿ, where n is a positive integer.
In this case, a = 1 and b = i, so we have (1 + i)⁸.
Using the binomial theorem, the expansion of (1 + i)⁸ is:
(1 + i)⁸ = C(8, 0) * 1⁸ * i⁰ + C(8, 1) * 1⁷ * i¹ + C(8, 2) * 1⁶ * i² + C(8, 3) * 1⁵ * i³ + C(8, 4) * 1⁴ * i⁴ + C(8, 5) * 1³ * i⁵ + C(8, 6) * 1² * i⁶ + C(8, 7) * 1¹ * i⁷ + C(8, 8) * 1⁰ * i⁸
Where C(n, r) represents the binomial coefficient, calculated as n! / (r! * (n - r)!)
Simplifying the above expression, we have:
(1 + i)⁸ = 1⁸ * i⁰ + 8 * 1⁷ * i¹ + 28 * 1⁶ * i² + 56 * 1⁵ * i³ + 70 * 1⁴ * i⁴ + 56 * 1³ * i⁵ + 28 * 1² * i⁶ + 8 * 1¹ * i⁷ + 1⁰ * i⁸
Since i² = -1, i³ = -i, i⁴ = 1, i⁵ = i, i⁶ = -1, i⁷ = -i, and i⁸ = 1, we can simplify further:
(1 + i)⁸ = 1 * 1 + 8 * 1 * i - 28 * 1 * 1 - 56 * 1 * i + 70 * 1 * 1 + 56 * 1 * i - 28 * 1 * 1 - 8 * 1 * i + 1 * 1
Simplifying the above expression, we get:
(1 + i)⁸ = 1 - 8i - 28 + 56i + 70 - 56i - 28 + 8i + 1
Combining like terms, we have:
(1 + i)⁸ = 64
Therefore, the value of (1 + i)⁸ is 64.